Abstract
We construct a class of continuous quasi-distances in a product of metric spaces and show that, generally, when the parameterλ(as shown in the paper) is positive,dis a distance and whenλ<0,dis only a continuous quasi-distance, but not a distance. It is remarkable that the same result in relation to the sign ofλwas found for two other classes of continuous quasi-distances (see Peppo (2010a, 2010b) and Peppo (2011)). This conclusion is due to the fact thatEis a product space. For the purposes of our main result, a notion of density in metric spaces is introduced.
Highlights
In this paper a quasi-distance d on a set E is defined as a function d : E2 → [0; +∞[ with the usual properties of a metric and a weaker version of the triangle inequality: d (x, y) ≤ k [d (x, z) + d (y, z)], (1)
This function is not always continuous with respect to the d-topology generated by itself in the same manner as by a distance. This is because the “open” balls B(a, r) = {x ∈ E : d(x, a) < r}, which form a base for a complete system of neighbourhoods of a ∈ E, are not always open sets in the dtopology
In [5,6,7] we proved that, generally, the functions defined in a product space for two points, P(x1, x2, . . . , xn) and Q(y1, y2, . . . , yn), respectively, by n d (P, Q)
Summary
In this paper a quasi-distance d on a set E is defined as a function d : E2 → [0; +∞[ with the usual properties of a metric and a weaker version of the triangle inequality:. This function is not always continuous with respect to the d-topology generated by itself in the same manner as by a distance. This is because the “open” balls B(a, r) = {x ∈ E : d(x, a) < r}, which form a base for a complete system of neighbourhoods of a ∈ E, are not always open sets in the dtopology.
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